Linear spaces of quadrics and new good codes
Let \(V\) be a vector space over the finite field \(F_q\) and \(Q\) the space of all quadratic forms on \(V\). From a subspace \(G\) of \(Q\) a linear code can be constructed: each codeword is obtained by evaluating a quadratic form of \(G\) in fixed representatives of all points in the projective space \(PV\). In his PhD thesis [`Codes: their parameters and geometry' (1997; Zbl 0866.94024)], \textit{M. A. de Boer} constructs a family of subspaces of \(Q\) that result in codes with high minimum distance. Moreover some parameters of these codes are computed. Conjecture 4.2.10 of the thesis deals with the number of nonzero weights of some of the constructed codes. In the current paper Brouwer proves Conjecture 4.2.10 of the thesis by de Boer and presents a shorter proof for the results on the parameters of the codes. Brouwer ends with a discussion on how to improve on the codes, thus obtaining codes that have higher minimum distance than all previously known codes.
scientific article; zbMATH DE number 1266804